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In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem ), which proves the existence of a particular kind of object ...
In classical real analysis, one way to define a real number is as an equivalence class of Cauchy sequences of rational numbers.. In constructive mathematics, one way to construct a real number is as a function ƒ that takes a positive integer and outputs a rational ƒ(n), together with a function g that takes a positive integer n and outputs a positive integer g(n) such that
Element (mathematics) Ur-element; Singleton (mathematics) Simple theorems in the algebra of sets; Algebra of sets; Power set; Empty set; Non-empty set; Empty function; Universe (mathematics) Axiomatization; Axiomatic system. Axiom schema; Axiomatic method; Formal system; Mathematical proof. Direct proof; Reductio ad absurdum; Proof by ...
From the other direction, there has been considerable clarification of what constructive mathematics is—without the emergence of a 'master theory'. For example, according to Errett Bishop's definitions, the continuity of a function such as sin(x) should be proved as a constructive bound on the modulus of continuity, meaning that the existential content of the assertion of continuity is a ...
In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. [1] Formulas from a formal theory are "realized" by objects, known as "realizers", in a way that knowledge of the realizer gives knowledge about the truth of the formula.
For example, any theorem of classical propositional logic of the form has a proof consisting of an intuitionistic proof of followed by one application of double-negation elimination. Intuitionistic logic can thus be seen as a means of extending classical logic with constructive semantics.